Fixed-point Quantum Walk Search.
Lvzhou Li, Yongzhen Xu, Zhang Delong · arXiv (Cornell University) · 2021
n this paper, we consider the problem of searching a marked vertex in a graph based on quantum walks. A very large body of research exists on this issue, from both theoretical and experimental studies, yet the existing quantum walk-based search algorithms require knowing exactly how many marked vertices there are before starting the search process. In addition, even if the precise knowledge of the number of marked vertices is known, the success probability of these algorithms could shrink dramatically when the number of search steps is greater than the right one. For avoiding these defects, in this paper we present a new quantum walk-based search algorithm which need not know any prior information about the marked vertices, but still keeps a quadratic speedup over classical ones and ensures that the error is bounded by a tunable parameter $\epsilon$. More specifically, for an $N$-vertex complete bipartite graph with marked vertices but without knowing the number of marked vertices, if the number of search steps $h$ satisfies $h \geq \ln(\frac{2}{\sqrt{\epsilon}})\sqrt{N}$, then the algorithm will output a marked vertex with probability at least $ 1-\epsilon$ for any given $\epsilon\in (0,1]$. This feature leads to quantum search algorithms with stronger robustness and a wider scope of application.